Optimal. Leaf size=24 \[ e^{-2 e} \left (4-x+\frac {1}{3} \left (-4+\frac {3}{(2+x)^2}\right )\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 20, normalized size of antiderivative = 0.83, number of steps used = 3, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {12, 2074} \begin {gather*} \frac {e^{-2 e}}{(x+2)^2}-e^{-2 e} x \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 2074
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=e^{-2 e} \int \frac {-10-12 x-6 x^2-x^3}{8+12 x+6 x^2+x^3} \, dx\\ &=e^{-2 e} \int \left (-1-\frac {2}{(2+x)^3}\right ) \, dx\\ &=-e^{-2 e} x+\frac {e^{-2 e}}{(2+x)^2}\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 16, normalized size = 0.67 \begin {gather*} -e^{-2 e} \left (x-\frac {1}{(2+x)^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.65, size = 30, normalized size = 1.25 \begin {gather*} -\frac {{\left (x^{3} + 4 \, x^{2} + 4 \, x - 1\right )} e^{\left (-2 \, e\right )}}{x^{2} + 4 \, x + 4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.12, size = 16, normalized size = 0.67 \begin {gather*} -{\left (x - \frac {1}{{\left (x + 2\right )}^{2}}\right )} e^{\left (-2 \, e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.04, size = 16, normalized size = 0.67
method | result | size |
default | \({\mathrm e}^{-2 \,{\mathrm e}} \left (-x +\frac {1}{\left (2+x \right )^{2}}\right )\) | \(16\) |
gosper | \(-\frac {\left (x^{3}-12 x -17\right ) {\mathrm e}^{-2 \,{\mathrm e}}}{x^{2}+4 x +4}\) | \(26\) |
risch | \(-{\mathrm e}^{-2 \,{\mathrm e}} x +\frac {{\mathrm e}^{-2 \,{\mathrm e}}}{x^{2}+4 x +4}\) | \(26\) |
norman | \(\frac {\left (12 \,{\mathrm e}^{-{\mathrm e}} x -{\mathrm e}^{-{\mathrm e}} x^{3}+17 \,{\mathrm e}^{-{\mathrm e}}\right ) {\mathrm e}^{-{\mathrm e}}}{\left (2+x \right )^{2}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.37, size = 21, normalized size = 0.88 \begin {gather*} -{\left (x - \frac {1}{x^{2} + 4 \, x + 4}\right )} e^{\left (-2 \, e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.06, size = 20, normalized size = 0.83 \begin {gather*} \frac {{\mathrm {e}}^{-2\,\mathrm {e}}}{{\left (x+2\right )}^2}-x\,{\mathrm {e}}^{-2\,\mathrm {e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [B] time = 0.16, size = 37, normalized size = 1.54 \begin {gather*} - \frac {x}{e^{2 e}} + \frac {1}{x^{2} e^{2 e} + 4 x e^{2 e} + 4 e^{2 e}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________